I.2.8

2/9/2021

Welll... Might as well quickly crank out another one before I go to bed. It's yet another projectivization of affine
question. Here's the affine version:

And unlike some of the trickier exercises like the last two, we can just follow the affine proof for this one. Here's
Theorem 1.11A, which we'll use:

And also Theorem 1.8A:

Since Y is a variety, note that I(Y ) is prime. And here's my slick proof:

dim Y | = n - 1 | |||||

⇐⇒ dim Y + 1 | = n | (Adding 1 to both sides) | ||||

⇐⇒ dim S(Y ) | = n | (by 2.6) | ||||

⇐⇒ dim k[x_{0},…,x_{n}] - htI(Y ) | = n | (by 1.8A) | ||||

⇐⇒n + 1 - htI(Y ) | = n | |||||

⇐⇒htI(Y ) | = 1 | |||||

⇐⇒I(Y ) | = (f) | where f satisfies the requirements |